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Lectures on Infinite Dimensional Lie Algebras

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Lectures on Infinite Dimensional Lie Algebras Lectures on Infinite Dimensional Lie Algebras Alexander Kleshchev Contents Part one: Kac-Moody Algebras page 1 1 Main Definitions 3 1.1 Some Examples 3 1.1.1 Special Linear Lie Algebras 3 1.1.2 Symplectic Lie Algebras 4 1.1.3 Orthogonal Lie Algebras 7 1.2 Generalized Cartan Matrices 10 1.3 The Lie algebra g˜(A) 13 1.4 The Lie algebra g(A) 16 1.5 Examples 20 2 Invariant bilinear form and generalized Casimir operator 26 2.1 Symmetrizable GCMs 26 2.2 Invariant bilinear form on g 27 2.3 Generalized Casimir operator 32 3 Integrable representations of g and the Weyl group 37 3.1 Integrable modules 37 3.2 Weyl group 39 3.3 Weyl group as a Coxeter group 42 3.4 Geometric properties of Weyl groups 46 4 The Classification of Generalized Cartan Matrices 50 4.1 A trichotomy for indecomposable GCMs 50 4.2 Indecomposable symmetrizable GCMs 58 4.3 The classification of finite and affine GCMs 61 5 Real and Imaginary Roots 68 5.1 Real roots 68 5.2 Real roots for finite and affine types 70 5.3 Imaginary roots 73 iii iv Contents 6 Affine Algebras 77 6.1 Notation 77 6.2 Standard bilinear form 77 6.3 Roots of affine algebra 80 6.4 Affine Weyl Group 84 6.4.1 A` 89 6.4.2 D` 89 6.4.3 E8 89 6.4.4 E7 90 6.4.5 E6 90 7 Affine Algebras as Central extensions of Loop Algebras 91 7.1 Loop Algebras 91 7.2 Realization of untwisted algebras 92 7.3 Explicit Construction of Finite Dimensional Lie Algebras 96 8 Twisted Affine Algebras and Automorphisms of Finite Order 99 8.1 Graph Automorphisms 99 8.2 Construction of Twisted Affine Algebras 108 8.3 Finite Order Automorphisms 114 9 Highest weight modules over Kac-Moody algebras 116 9.1 The category O 116 9.2 Formal Characters 118 9.3 Generators and relations 122 10 Weyl-Kac Character formula 127 10.1 Integrable highest weight modules and Weyl group 127 10.2 The character formula 128 10.3 Example: Lˆ(sl2) 132 10.4 Complete reducibility 134 10.5 Macdonald’s identities 136 10.6 Specializations of Macdonald’s identities 139 10.7 On converegence of characters 141 11 Irreducible Modules for affine algebras 144 11.1 Weights of irreducible modules 144 11.2 The fundamental modules for sbl2 151 Bibliography 155 Part one Kac-Moody Algebras 1 Main Definitions 1.1 Some Examples 1.1.1 Special Linear Lie Algebras Let g = sln = sln(C). Choose the subalgebra h consisting of all diagonal ∨ matrices in g. Then, setting αi := eii − ei+1,i+1, ∨ ∨ α1 , . , αn−1 ∗ is a basis of h. Next define ε1 . , εn ∈ h by εi : diag(a1, . , an) 7→ ai. Then, setting αi = εi − εi+1, α1, . , αn−1 is a basis of h∗. Let ∨ aij = hαi , αji. Then the (n − 1) × (n − 1) matrix A := (aij) is 2 −1 0 0 ... 0 0 0 −1 2 −1 0 ... 0 0 0 0 −1 2 −1 ... 0 0 0 . . . . 0 0 0 0 ... −1 2 −1 0 0 0 0 ... 0 −1 2 This matrix is called the Cartan matrix. Define Xεi−εj := eij,X−εi+εj := eji (1 ≤ i < j ≤ n) 3 4 Main Definitions Note that [h, Xα] = α(h)Xα (h ∈ h), and ∨ ∨ {α1 , . , αn−1} ∪ {Xεi−εj | 1 ≤ i 6= j ≤ n} is a basis of g. Set ei = Xαi and fi = X−αi for 1 ≤ i < n. It is easy to check that ∨ ∨ e1, . , en−1, f1, . , fn−1, α1 , . αn−1 (1.1) generate g and the following relations hold: ∨ [ei, fj] = δijαi , (1.2) ∨ ∨ [αi , αj ] = 0, (1.3) ∨ [αi , ej] = aijej, (1.4) ∨ [αi , fj] = −aijfj, (1.5) 1−aij (ad ei) (ej) = 0 (i 6= j), (1.6) 1−aij (ad fi) (fj) = 0 (i 6= j). (1.7) A (special case of a) theorem of Serre claims that g is actually gener- ated by the elements of (1.1) subject only to these relations. What is important for us is the fact that the Cartan matrix contains all the in- formation needed to write down the Serre’s presentation of A. Since the Cartan matrix is all the data we need, it makes sense to find a nicer ge- ometric way to picture the same data. Such picture is called the Dynkin diagram, and in our case it is: ••••••••... α1 α2 αn−1 Here vertices i and i + 1 are connected because ai,i+1 = ai+1,i = −1, others are not connected because aij = 0 for |i − j| > 1, and we don’t have to record aii since it is always going to be 2. 1.1.2 Symplectic Lie Algebras Let V be a 2n-dimensional vector space and ϕ : V × V → C be a non-degenerate symplectic bilinear form on V . Let g = sp(V, ϕ) = {X ∈ gl(V ) | ϕ(Xv, w) + ϕ(v, Xw) = 0 for all v, w ∈ V }. 1.1 Some Examples 5 An easy check shows that g is a Lie subalgebra of gl(V ). It is known from linear algebra that over C all non-degenerate symplectic forms are equivalent, i.e. if ϕ0 is another such form then ϕ0(v, w) = ϕ(gv, gw) for some fixed g ∈ GL(V ). It follows that sp(V, ϕ0) = g−1(sp(V, ϕ))g =∼ sp(V, ϕ), thus we can speak of just sp(V ). To think of sp(V ) as a Lie algebra of matrices, choose a symplectic basis e1, . , en, e−n, . , e−1, that is ϕ(ei, e−i) = −ϕ(e−i, ei) = 1, and all other ϕ(ei, ej) = 0. Then the Gram matrix is 0 s G = , −s 0 where 0 0 ... 0 1 0 0 ... 1 0 . s = . . (1.8) 0 1 ... 0 0 1 0 ... 0 0 It follows that the matrices of sp(V ) in the basis of ei’s are precisely the matrices from the Lie algebra AB sp = { | B = sBts, C = sCts, D = −sAts}, 2n CD ∼ t so sp(V ) = sp2n. Note that sX s is the transpose of X with respect to the second main diagonal. Choose the subalgebra h consisting of all diagonal matrices in g. Then, ∨ setting αi := eii − ei+1,i+1 − e−i,−i + e−i−1,−i−1, for 1 ≤ i < n and ∨ αn = enn − e−n,−n, ∨ ∨ ∨ α1 , . , αn−1, αn is a basis of h. Next, setting αi = εi −εi+1 for 1 ≤ i < n, and αn := 2εn, α1, . , αn−1, αn is a basis of h∗. Let ∨ aij = hαi , αji. 6 Main Definitions Then the Cartan matrix A is the n × n matrix 2 −1 0 0 ... 0 0 0 −1 2 −1 0 ... 0 0 0 0 −1 2 −1 ... 0 0 0 . . . . 0 0 0 0 ... −1 2 −2 0 0 0 0 ... 0 −1 2 Define X2εi = ei,−i, (1 ≤ i ≤ n) X−2εi = e−i,i, (1 ≤ i ≤ n) Xεi−εj = eij − e−j,−i (1 ≤ i < j ≤ n) X−εi+εj = eji − e−i,−j (1 ≤ i < j ≤ n) Xεi+εj = ei,−j + ej,−i (1 ≤ i < j ≤ n) X−εi−εj = e−j,i + e−i,j (1 ≤ i < j ≤ n). Note that [h, Xα] = α(h)Xα (h ∈ h), and ∨ ∨ {α1 , . , αn } ∪ {Xα} is a basis of g. Set ei = Xαi and fi = X−αi for 1 ≤ i ≤ n. It is easy to check that ∨ ∨ e1, . , en, f1, . , fn, α1 , . αn (1.9) generate g and the relations (1.2-1.7) hold. Again, Serre’s theorem claims that g is actually generated by the elements of (1.11) subject only to these relations. The Dynkin diagram in this case is: ••••••••... < α1 α2 αn−1 αn The vertices n−1 and n are connected the way they are because an−1,n = −2 and an,n−1 = −1, and in other places we follow the same rules as in the case sl. 1.1 Some Examples 7 1.1.3 Orthogonal Lie Algebras Let V be an N-dimensional vector space and ϕ : V × V → C be a non-degenerate symmetric bilinear form on V . Let g = so(V, ϕ) = {X ∈ gl(V ) | ϕ(Xv, w) + ϕ(v, Xw) = 0 for all v, w ∈ V }. An easy check shows that g is a Lie subalgebra of gl(V ). It is known from linear algebra that over C all non-degenerate symmetric bilinear forms are equivalent, i.e. if ϕ0 is another such form then ϕ0(v, w) = ϕ(gv, gw) for some fixed g ∈ GL(V ). It follows that so(V, ϕ0) = g−1(so(V, ϕ))g =∼ so(V, ϕ), thus we can speak of just so(V ). To think of so(V ) as a Lie alge- bra of matrices, choose a basis e1, . , en, e−n, . , e−1 if N = 2n and e1, . , en, e0, e−n, . , e−1 if N = 2n + 1, such that the Gram matrix of ϕ in this basis is 0 0 s 0 s and 0 2 0 , s 0 s 0 0 respectively, where s is the n × n matrix as in (1.8). It follows that the matrices of so(V ) in the basis of ei’s are precisely the matrices from the Lie algebra AB so = { | B = −sBts, C = −sCts, D = −sAts}, 2n CD if N = 2n, and A 2sxt B t t t so2n+1 = { y 0 x | B = −sB s, C = −sC s, D = −sA s}, C 2syt D if N = 2n + 1 (here x, y are arbitrary 1 × n matrices).
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